Some Aspects Of Nonlinearized Eigenvalue Problem For Kaup-Newell System

Based on the 2×2 Kaup-Newell spectral problem, A 3 x 3 eigenvalue problem is proposed with the help of adjoint representation. And the relations between them are also given. This paper explores the way of Lie-Poisson system for nonlinearied eigenvalue problems applying to soliton equations. Further, the general form of 2 x 2 nonlinearized eigenvalue problems is presented. Reduction of the Lie-Poisson structure on the set of Mλyields the Rosochatius system and the standard Hamilton system. Finally, the separation equations and the straighten flows are obtained by using Hamilton-Jacobi theory.